Short answers, pulled from the story.
Bose-Einstein statistics was developed by Satyendra Nath Bose and Albert Einstein between 1924 and 1925. Bose introduced the framework for photons in 1924, and Einstein generalized it to atoms in 1924-25.
Bose-Einstein statistics explains the cohesive streaming of laser light, the frictionless creeping of superfluid helium, and the existence of the Bose-Einstein condensate. The condensate, a dense collection of bosons all in the same ground state, was demonstrated experimentally in 1995.
Bose made an accidental error while lecturing at the University of Dhaka, treating photons as indistinguishable from one another. The error produced a result that agreed with experimental data, leading Bose to recognize that Maxwell-Boltzmann statistics did not apply to all microscopic particles.
Einstein personally translated Bose's article from English into German after Bose sent the rejected manuscript to him requesting publication in the Zeitschrift fur Physik. Einstein then published his own supporting paper alongside it in 1924.
Bosons have integer values of spin and are not restricted by the Pauli exclusion principle, so any number of them can occupy the same quantum state. Fermions have half-integer spins and obey Fermi-Dirac statistics, which limits them to one particle per state.
Bose-Einstein statistics has been applied to information retrieval as a term-weighting model under the Divergence From Randomness framework, with source code available from the Terrier project at the University of Glasgow. It has also been used to analyze complex networks such as the World Wide Web, where it predicts that first-mover advantage and winner-takes-all outcomes are thermodynamically distinct phases.